FORMATION OF ORTHOGONAL GRIDS ON THE BASIS OF FUNDAMENTAL SPLINES

Abstract

To model smooth curved contours the questions are raised about finding derivatives that can be formed based on user additional information. If such information does not exist, then additional numerical methods are used for calculation. To avoid extra calculations, it is relevant to use fundamental splines which are based only on the point frame information.To work with zero-length curves that allows the formation of orthogonal grids on a plane, additional research is needed in order to save this condition for different values ​​of parameter replacement.

The author proposes a method of constructing flat orthogonal grids based on a flat isotropic fundamental spline. The conditions for creating a three-dimensional fundamental spline are given. To form a flat isotropic spline, the condition of isotropic chords is used, which subtend the point frame and forms the statement. It is proposed to use a quasi-conformal parameter change to form an orthogonal grid. The parameter is replaced by a combination of the defined functions f₁(u), f₂(v) - where u and v are some valid parameters. The functions must be formed on the basis of different parameters. The statement is given regarding the formation of an orthogonal grid based on an isotropic spline with the proposed parameter replacement. To prove the validity of the proposed states, examples of simulated grids are given and the coefficient of the first quadratic form F is calculated, the analysis of which allows us to make certain assumptions. Further studies are related to the construction of minimal surfaces based on the proposed quasi-conformal replacement.

Keywords: isotropic curve, fundamental spline, orthogonal grid, isothermal grid, quasiconformal parameter replacement.